Preamble

This section addresses the solution of a specific class of partial differential equations with time delay. We consider the following equation:

$$ \frac{\partial^2 u(t, x)}{\partial t^2} = a^2 \frac{\partial^2 u(t, x)}{\partial x^2} - b^2 \frac{\partial^2 u(t - \tau, x)}{\partial x^2} + f(t, x) \tag{1} $$

where $a^2 > b^2 > 0$ and $\tau > 0$ represents the time delay. The system is subject to the following boundary conditions:

$$ u(t, 0) = 0, \quad u(t, l) = 0 \tag{2} $$

To solve this problem, we employ the method of separation of variables by assuming a solution of the form $u(t, x) = T(t)X(x)$. Substituting this into the homogeneous part of equation (1), we obtain the spatial eigenvalue problem:

$$ X''(x) + \lambda X(x) = 0, \quad X(0) = X(l) = 0 \tag{3} $$

The eigenvalues and corresponding eigenfunctions are given by:

$$ \lambda_n = \frac{n^2 \pi^2}{l^2}, \quad X_n(x) = \sin \frac{n \pi x}{l}, \quad (n = 1, 2, \dots) \tag{4} $$

The general solution can be represented as a series:

$$ u(t, x) = \sum_{n=1}^{\infty} T_n(t) \sin \frac{n \pi x}{l} \tag{5} $$

Substituting (5) into (1), we derive a sequence of ordinary differential equations for the time-dependent coefficients $T_n(t)$:

$$ T_n''(t) + a^2 \lambda_n T_n(t) - b^2 \lambda_n T_n(t - \tau) = f_n(t) \tag{6} $$

where $f_n(t)$ are the Fourier coefficients of the source term $f(t, x)$. For the initial interval $0 \le t \le \tau$, the function $T_n(t)$ is determined by the initial conditions. Let $B_n(t)$ represent the source term contribution. Using the method of successive integration over intervals of length $\tau$, we can express the solution for $T_n(t)$ as:

$$ T_n(t) = C_{0n} + C_{1n}t + C_{2n}t^2 + \int_0^t T_n(t-s) y_n(s) ds \tag{7} $$

The coefficients $C_{0n}, C_{1n}, C_{2n}$ are determined by the continuity requirements at the boundaries of the time intervals. Specifically, for $t \in [k\tau, (k+1)\tau]$, the solution depends on the values of the function in the preceding interval $[(k-1)\tau, k\tau]$. By applying the properties of the integral kernel and the specific form of $B_n(t)$, we obtain:

$$ C_{0n} = T_n(0) + \frac{\tau}{2} T_n'(\tau) - \dots \tag{8} $$

The final solution for the displacement $u(t, x)$ is constructed by summing the components:

$$ u(t, x) = \sum_{n=1}^{\infty} \left{ C_{0n} T_n(t) + C_{1n} T_n(t) + \int_0^t T_n(t-s) \left[ h_n(s) - \int_0^s B_n(\xi) \cos \omega(s-\xi) d\xi \right] ds \right} \sin \frac{n \pi x}{l} \tag{9} $$

Estimates for the coefficients and the integral terms show that for $t \in [k\tau, (k+1)\tau]$, the following inequality holds:

$$ |T_n(t)|, |T_n'(t)|, |T_n''(t)| < n^{k-1} C_k \tag{10} $$

where $C_k$ is a constant depending on the interval index. These bounds ensure the convergence of the series (9) and its derivatives, provided the source function $f(t, x)$ and the initial data possess sufficient smoothness (typically up to order $k+4$).

As a practical example, consider the case where $l=1, \tau=1$, and the source term is given by $f(t, x) = x^6(x-1)^6 \sin t$. The solution $u(t, x)$ can be computed numerically or analytically using the derived formulas for successive time intervals. The results demonstrate the influence of the delay parameter $\tau$ on the wave propagation characteristics within the medium.